Distinct cubic fields conjecture for the initial point sets $I_{0,c}$

Let $I_{0,c}$ be the initial point set associated with the positive integer $c$, whose elements are cubic algebraic integers. Distinct cubic fields conjecture. For every $c\in\mathbb{Z}_{>0}$ and all distinct $\alpha,\beta\in I_{0,c}$, one has

$$\mathbb{Q}(\alpha)\neq\mathbb{Q}(\beta).$$

Equivalently, each element of $I_{0,c}$ belongs to a different cubic number field. The claim is supported by the authors' computational experiments for $1\leq c\leq 5\times 10^4$, but no proof or resolution is given.